Yes I've had another look - the dates certainly match up - he (FLL) was born circa 1811-1812 from age on convict records, which matches the baptism. Exeter, Devon matches.
I haven't been able to find any evidence of a death of a Frederick Augustus Lindsay/Lindsey etc or any other reference to him on census records, though he could have died before they were taken and the death not recorded. Still, there's no record there to prove there WAS a Frederick Augustus Lindsey survived in England AFTER my fella came out here. Or even a Frederick Lindsey of the right age and location that was really obvious...
I think there's a good chance you've found his parents for me

Just have to say "possible parents" rather than "parents" in adding them to my tree.
As to what became of him - I looked further into the death of Frederick Lindsay in Victoria in 1854. He was 42 years of age (matches b1812), parents were unknown, but birthplace was England. I think I might have found my man! Of course this could all be co-incidental, but I'm excited just a little

Notwithstanding the plethora of spelling variations from Lindsey at baptism to Lindsay at death and all the variations in-between of Linsey, Linsay etc - plus changing his middle name from Augustus to Lillycroft/Lillycraft/Lillycropp/Lillycrapp etc.
I still haven't found the story behind the Lee name however - it would appear that Frederick the father didn't change his name to Lee, so that is one thing that can perhaps be eliminated. I also don't know how Agnes as a convict had an illegitimate baby and didn't seem to suffer the consequences for it (ie convict women generally got a further punishment if they were found to be pregnant, irrespective of the circumstances). Agnes would have conceived around April 1840 to have Frederick in Jan 1841. There is nothing in her conduct record in 1840 until October when she got done for "insolence". She would have been 6 months pregnant by then - maybe she managed to keep it hidden till that time...?? but there is no evidence of Frederick at the Queen's orphanage which is where the convict babies ended up as a rule. Agnes was working at the factory nursery - maybe she managed to keep him hidden amongst the babies there....otherwise I can only imagine she somehow had someone else looking after him on the quiet on the outside - I'm wondering if the Lee name is in thanks for someone of that name who took care of him....

?
Still a puzzle!
P.S. Frederick junior was known as Frederick Lee on all documents including electoral rolls up until about 9 years before his death. In the last few years he appears as Frederick Leslie Lee on the roll, and also named Frederick Leslie Lee on his death certificate. From where and why this middle name appeared is also a mystery!